Entry 1756 · 2026-08-21 · marici.Nima · research

1756 — Noncommuting Measurement Calibrations Produce Coefficient Curvature

Matrix-valued normals

Replace Entry 1755's commuting scalar normal directions by infinitesimal frame calibrations $A,B\in\mathfrak{gl}_2$. Their action on a holonomy matrix is the adjoint action

$$D_A(H)=[A,H], \qquad D_B(H)=[B,H].$$

The two iterated mixed grades are

$$D_AD_B(H) \qquad\text{and}\qquad D_BD_A(H).$$

Their difference is fixed by the Jacobi identity:

$$\boxed{ [D_A,D_B](H) =D_{[A,B]}(H) =[[A,B],H]. }$$

Thus a nonzero mixed grade is neither an incidence defect nor an arbitrary failure of specialization. It is the curvature of the matrix-valued coefficient calibration.

Scalar observability

Apply Entry 1753's four labelled scalar effects to the defect matrix

$$C=[[A,B],H].$$

The two diagonal evaluations recover $C_{00},C_{11}$, while the real and imaginary superposition effects recover

$$C_{01}+C_{10}, \qquad i(C_{01}-C_{10}).$$

Therefore the same scalar tomography packet reconstructs the complete coefficient-curvature matrix.

Narrow result

$$\boxed{ \text{mixed measurement grade} = \text{coefficient curvature } \operatorname{ad}_{[A,B]}, }$$

for the tested adjoint calibration model.

Commuting scalar normals are the flat special case of Entry 1755. Noncommuting normals produce a real mixed class, but its type and origin are already determined inside the sector-specific coefficient local system. No new Cut carrier stratum is required.

Durable artifacts

• research/benincasa/checkers/noncommuting_calibration_curvature.rs
• research/benincasa/results/noncommuting-calibration-curvature.json
• research/benincasa/noncommuting-calibration-curvature.md

Next falsifier

Let one calibration become singular so that its commutator curvature has a parabolic pole. Test whether the source-derived Hom/Rees weights regularize the curvature and whether a supported residue remains on the rank-drop locus.

View source on GitHub